Preheat an indoor or outdoor grill to high. Bring 1/2 inch of water to a boil in a medium-size pot.
In a bowl, combine the ground beef and pork with the fennel seeds, garlic, parsley, crushed red pepper flakes, Parmigiano, grill seasoning and 1 tablespoon of EVOO. With the side of your hand make an X indentation in the bowl dividing the mixture into 4 equal portions. Form each portion into a patties about one-inch-thick. Drizzle them with some EVOO, transfer to the grill and cook about 6-8 minutes on each side.
Place a large saute pan over medium-high heat, add 2 turns of the pan EVOO, about 2 tablespoons. Add the onions, bell and Cubanelle peppers, and season with salt and pepper. Cook, stirring every now and then, until the onions and peppers are tender, about 8-10 minutes.
While the burgers are cooking, add a generous amount of salt to the boiling water. Add the green beans and put a lid on it. Steam them for 3-4 minutes. Cold shock the beans by running them under cold water then drain them well. Place beans in a bowl and combine with the red onion, cucumber and tomato. Dress salad with a generous drizzle of EVOO and season generously with salt and freshly ground black pepper. Add the balsamic vinegar and toss to coat.
Split the rolls and toast briefly on the grill, until just marked. Place a patty on a roll, top with 2 slices of chees and the onion pepper mixture, and serve with green bean salad alongside.